Are you treating the arguments in the assertions as independent random variables? Because the same symbol will show up in several assertions, in which case the meanings aren’t independent.
It makes what you did a first approximation. But I think you can go further. Call S the set of all s and P the set of all p. Each proposition links a p in P to some subset of S. A set of such propositions can be represented by some sort of hypergraph built on S and P. So there’s a hypergraph for g(K), another hypergraph for W, and you want to estimate the number of sub-hypergraphs of W which are isomorphic to g(K). It’s complicated but it may be doable, especially if you can find a way to make that estimate given just statistical properties of the hypergraphs.
I doubt that thinking of the arguments as forming a subset of S is the end, because predicates typically have more structure than just their arity. E.g. the order of arguments may or may not matter. So the sets need to be ordered, or the links from P to S need to be labeled so as to indicate the logical form of the predicate. But one step at a time!
A predicate logic representation is a hypergraph on P and S. g(K) maps that hypergraph into a hypergraph on W, kind of (but K is intensional and W is extensional, so simple examples using English-level concepts will run into trouble).
When you ask about finding sub-hypergraphs of W that are isomorphic to g(K), it sounds like you’re just asking if g(K) evaluates to true.
I just skimmed your definitions so I might be getting something wrong. But the idea is that your symbolic reasoning system, prior to interpretation, corresponds to a system of propositional schemas, in which the logical form and logical dependencies of the propositions have been specified, but the semantics has not. Meanwhile, the facts about the world are described by another, bigger system of propositions (whose semantics is fixed). The search for groundings is the search for subsystems in this description of the world which have the logical structure (form and dependency) of those in the uninterpreted schema-system. The stuff about hypergraphs was just a guess as to the appropriate mathematical framework. But maybe posets or lattices are appropriate (for the deductive structure).
I haven’t thought about intension vs extension at all. That might require a whole other dimension of structure.
Are you treating the arguments in the assertions as independent random variables? Because the same symbol will show up in several assertions, in which case the meanings aren’t independent.
I think you mean that the probability of multiple predicates involving the same variable being true under a different grounding are not independent.
I hadn’t thought of that. That could be devastating.
It makes what you did a first approximation. But I think you can go further. Call S the set of all s and P the set of all p. Each proposition links a p in P to some subset of S. A set of such propositions can be represented by some sort of hypergraph built on S and P. So there’s a hypergraph for g(K), another hypergraph for W, and you want to estimate the number of sub-hypergraphs of W which are isomorphic to g(K). It’s complicated but it may be doable, especially if you can find a way to make that estimate given just statistical properties of the hypergraphs.
I doubt that thinking of the arguments as forming a subset of S is the end, because predicates typically have more structure than just their arity. E.g. the order of arguments may or may not matter. So the sets need to be ordered, or the links from P to S need to be labeled so as to indicate the logical form of the predicate. But one step at a time!
I don’t follow.
A predicate logic representation is a hypergraph on P and S. g(K) maps that hypergraph into a hypergraph on W, kind of (but K is intensional and W is extensional, so simple examples using English-level concepts will run into trouble).
When you ask about finding sub-hypergraphs of W that are isomorphic to g(K), it sounds like you’re just asking if g(K) evaluates to true.
I just skimmed your definitions so I might be getting something wrong. But the idea is that your symbolic reasoning system, prior to interpretation, corresponds to a system of propositional schemas, in which the logical form and logical dependencies of the propositions have been specified, but the semantics has not. Meanwhile, the facts about the world are described by another, bigger system of propositions (whose semantics is fixed). The search for groundings is the search for subsystems in this description of the world which have the logical structure (form and dependency) of those in the uninterpreted schema-system. The stuff about hypergraphs was just a guess as to the appropriate mathematical framework. But maybe posets or lattices are appropriate (for the deductive structure).
I haven’t thought about intension vs extension at all. That might require a whole other dimension of structure.